Engineering Optimization

39Case Study 4: A Goddard Rocket Study

39.1 The Process and Analysis

Yes, it is rocket science; although simplified here, it is adequately rich. The objective is to schedule thrust (e.g., change thrust with time) to maximize the altitude a rocket can achieve. You could have full thrust until the fuel burns out, and then let the rocket coast up until gravity pulls it back. But this plan makes the rocket go fast in the lower altitude where the air resistance is high and wastes energy. A better plan to minimize the impact of air resistance is to move slowly in the low altitudes and then full throttle to fast speed in the high altitudes. But moving slow in the low altitudes, in the high gravitational field, means wasting fuel to fight gravity. In a limit of not enough thrust to overcome gravity, the rocket never rises; it burns all its fuel on the launch pad. This optimization is known as the “Goddard problem” in honor of rocket scientist Robert Goddard.

This will consider a simple situation of a single‐stage vertical rocket (no changes in drag coefficient), moving vertically from a stationary flat earth (no Coriolis effects) through an atmosphere with nominal density dependence on height. Even with such idealizations, it will be sufficiently complex!

One generally accepted solution is to accelerate the rocket at full thrust to terminal velocity (the speed at which it would fall if pulled down by gravity when air drag resistance exactly balanced gravity) and then reduce thrust ...